Numerical Methods and Algorithms

By: Alan M. Cohen

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Hardcover | 15 August 2007

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268 Pages

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Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations. When solving such problems, in many cases it is fairly easy to obtain the Laplace transform, while it is very demanding to determine the inverse Laplace transform that is the solution of a given problem. Sometimes, after some difficult contour integration, we may find that a series solution results, but this may be quite difficult to evaluate in order to get an answer at a particular time value.

The advent of computers has given an impetus to developing numerical methods for the determination of the inverse Laplace transform. This book gives background material on the theory of Laplace transforms, together with a fairly comprehensive list of methods that are available at the current time. Computer programs are included for those methods that perform consistently well on a wide range of Laplace transforms.

Industry Reviews

From the reviews:

"This book gives background material on the theory of Laplace transforms together with a comprehensive list of numerical methods for determination of the inverse Laplace transform. Computer programs are included for those methods that perform consistently well on a wide range of Laplace transforms. It provides a balanced course of Laplace transforms consisting of theory, numerical techniques and applications. ... This book may be a useful tool for all those who use Laplace transforms in their work whether they are mathematicians, engineers or financial planners." (Som Prakash Goyal, Zentralblatt MATH, Vol. 1127 (4), 2008)

"In this book, Alan M. Cohen summarizes known results related to numerical methods. The book starts with basic analytic theory of the Laplace transform. ... The book provides a large amount of carefully selected and clearly explained examples, and seems to be ideal for self-study or preparing students' courses for different levels. By gathering various information on the Laplace transform in one place, the author has created a standard reference source." (Alexander Denisjuk, Mathematical Reviews, Issue 2009 c)

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Applied Mathematics Integral Calculus & Equations Engineering & Technology Technology in General Engineering in General Maths for Engineers

Preface	p. viii
Acknowledgements	p. xii
Notation	p. xiii
Basic Results	p. 1
Introduction	p. 1
Transforms of Elementary Functions	p. 2
Elementary Properties of Transforms	p. 3
Transforms of Derivatives and Integrals	p. 5
Inverse Transforms	p. 8
Convolution	p. 9
The Laplace Transforms of some Special Functions	p. 11
Difference Equations and Delay Differential Equations	p. 14
z-Transforms	p. 16
Multidimensional Laplace Transforms	p. 18
Inversion Formulae and Practical Results	p. 23
The Uniqueness Property	p. 23
The Bromwich Inversion Theorem	p. 26
The Post-Widder Inversion Formula	p. 37
Initial and Final Value Theorems	p. 39
Series and Asymptotic Expansions	p. 42
Parseval's Formulae	p. 43
The Method of Series Expansion	p. 45
Expansion as a Power Series	p. 45
An alternative treatment of series expansions	p. 49
Expansion in terms of Orthogonal Polynomials	p. 48
Legendre Polynomials	p. 50
Chebyshev Polynomials	p. 52
Laguerre Polynomials	p. 55
The method of Weeks	p. 58
Multi-dimensional Laplace transform inversion	p. 66
Quadrature Methods	p. 71
Interpolation and Gaussian type Formulae	p. 71
Evaluation of Trigonometric Integrals	p. 75
Extrapolation Methods	p. 77
The P-transformation of Levin	p. 77
The Sidi mW-Transformation for the Bromwich integral	p. 78
Methods using the Fast Fourier Transform (FFT)	p. 81
Hartley Transforms	p. 91
Dahlquist's "Multigrid" extension of FFT	p. 95
Inversion of two-dimensional transforms	p. 100
Rational Approximation Methods	p. 103
The Laplace Transform is Rational	p. 103
The least squares approach to rational Approximation	p. 106
Sidi's Window Function	p. 108
The Cohen-Levin Window Function	p. 109
Pade, Pade-type and Continued Fraction Approximations	p. 111
Prony's method and z-transforms	p. 116
The Method of Grundy	p. 118
Multidimensional Laplace Transforms	p. 119
The Method of Talbot	p. 121
Early Formulation	p. 121
A more general formulation	p. 123
Choice of Parameters	p. 125
Additional Practicalities	p. 129
Subsequent development of Talbot's method	p. 130
Piessens' method	p. 130
The Modification of Murli and Rizzardi	p. 132
Modifications of Evans et al	p. 133
The Parallel Talbot Algorithm	p. 137
Multi-precision Computation	p. 138
Methods based on the Post-Widder Inversion Formula	p. 141
Introduction	p. 141
Methods akin to Post-Widder	p. 143
Inversion of Two-dimensional Transforms	p. 146
The Method of Regularization	p. 147
Introduction	p. 147
Fredholm equations of the first kind - theoretical considerations	p. 148
The method of Regularization	p. 150
Application to Laplace Transforms	p. 151
Survey Results	p. 157
Cost's Survey	p. 157
The Survey by Davies and Martin	p. 158
Later Surveys	p. 160
Narayanan and Beskos	p. 160
Duffy	p. 161
D'Amore, Laccetti and Murli	p. 161
Cohen	p. 162
Test Transforms	p. 168
Applications	p. 169
Application 1. Transient solution for the Batch Service Queue M/M^2+N/1	p. 169
Application 2. Heat Conduction in a Rod	p. 178
Application 3. Laser Anemometry	p. 181
Application 4. Miscellaneous Quadratures	p. 187
Application 5. Asian Options	p. 191
Appendix	p. 197
Table of Laplace Transforms	p. 198
Table of z-Transforms	p. 203
The Fast Fourier Transform (FFT)	p. 204
Fast Hartley Transforms (FHT)	p. 206
Quadrature Rules	p. 206
Extrapolation Techniques	p. 212
Padé Approximation	p. 220
Continued Fractions. Thiele's method	p. 223
The method of Steepest Descent	p. 226
Gerschgorin's theorems and the Companion Matrix	p. 227
Bibliography	p. 231
Index	p. 248
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Numerical Methods and Algorithms

At a Glance

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